Precalculus Examples

$f (x) = 4 + \sqrt[3]{x}$

Step 1

Write $f (x) = 4 + \sqrt[3]{x}$ as an equation.

$y = 4 + \sqrt[3]{x}$

Step 2

Interchange the variables.

$x = 4 + \sqrt[3]{y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $4 + \sqrt[3]{y} = x$ .

$4 + \sqrt[3]{y} = x$

Step 3.2

Subtract $4$ from both sides of the equation.

$\sqrt[3]{y} = x - 4$

Step 3.3

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y}}^{3} = {(x - 4)}^{3}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

${(y^{\frac{1}{3}})}^{3} = {(x - 4)}^{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{3} \cdot 3} = {(x - 4)}^{3}$

Step 3.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(x - 4)}^{3}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(x - 4)}^{3}$

Step 3.4.2.1.2

Simplify.

$y = {(x - 4)}^{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x - 4)}^{3}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$y = x^{3} + 3 x^{2} \cdot - 4 + 3 x {(- 4)}^{2} + {(- 4)}^{3}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Multiply $- 4$ by $3$ .

$y = x^{3} - 12 x^{2} + 3 x {(- 4)}^{2} + {(- 4)}^{3}$

Step 3.4.3.1.2.2

Raise $- 4$ to the power of $2$ .

$y = x^{3} - 12 x^{2} + 3 x \cdot 16 + {(- 4)}^{3}$

Step 3.4.3.1.2.3

Multiply $16$ by $3$ .

$y = x^{3} - 12 x^{2} + 48 x + {(- 4)}^{3}$

Step 3.4.3.1.2.4

Raise $- 4$ to the power of $3$ .

$y = x^{3} - 12 x^{2} + 48 x - 64$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = x^{3} - 12 x^{2} + 48 x - 64$

Step 5

Verify if $f^{- 1} (x) = x^{3} - 12 x^{2} + 48 x - 64$ is the inverse of $f (x) = 4 + \sqrt[3]{x}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (4 + \sqrt[3]{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = {(4 + \sqrt[3]{x})}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (4 + \sqrt[3]{x}) = 4^{3} + 3 \cdot (4^{2} \sqrt[3]{x}) + 3 \cdot (4 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Raise $4$ to the power of $3$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 3 \cdot (4^{2} \sqrt[3]{x}) + 3 \cdot (4 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.2

Raise $4$ to the power of $2$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 3 \cdot (16 \sqrt[3]{x}) + 3 \cdot (4 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.3

Multiply $3$ by $16$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 3 \cdot (4 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.4

Multiply $3$ by $4$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 {\sqrt[3]{x}}^{2} + {\sqrt[3]{x}}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.5

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + {\sqrt[3]{x}}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.6

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 5.2.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + {(x^{\frac{1}{3}})}^{3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x^{\frac{1}{3} \cdot 3} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.6.3

Combine $\frac{1}{3}$ and $3$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x^{\frac{3}{3}} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.6.4

Cancel the common factor of $3$ .

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Step 5.2.3.2.6.4.1

Cancel the common factor.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x^{\frac{3}{3}} - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.6.4.2

Rewrite the expression.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.2.6.5

Simplify.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 {(4 + \sqrt[3]{x})}^{2} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.3

Rewrite ${(4 + \sqrt[3]{x})}^{2}$ as $(4 + \sqrt[3]{x}) (4 + \sqrt[3]{x})$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 ((4 + \sqrt[3]{x}) (4 + \sqrt[3]{x})) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.4

Expand $(4 + \sqrt[3]{x}) (4 + \sqrt[3]{x})$ using the FOIL Method.

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Step 5.2.3.4.1

Apply the distributive property.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (4 (4 + \sqrt[3]{x}) + \sqrt[3]{x} (4 + \sqrt[3]{x})) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.4.2

Apply the distributive property.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (4 \cdot 4 + 4 \sqrt[3]{x} + \sqrt[3]{x} (4 + \sqrt[3]{x})) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.4.3

Apply the distributive property.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (4 \cdot 4 + 4 \sqrt[3]{x} + \sqrt[3]{x} \cdot 4 + \sqrt[3]{x} \sqrt[3]{x}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.5

Simplify and combine like terms.

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Step 5.2.3.5.1

Simplify each term.

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Step 5.2.3.5.1.1

Multiply $4$ by $4$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (16 + 4 \sqrt[3]{x} + \sqrt[3]{x} \cdot 4 + \sqrt[3]{x} \sqrt[3]{x}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.5.1.2

Move $4$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (16 + 4 \sqrt[3]{x} + 4 \sqrt[3]{x} + \sqrt[3]{x} \sqrt[3]{x}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.5.1.3

Multiply $\sqrt[3]{x} \sqrt[3]{x}$ .

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Step 5.2.3.5.1.3.1

Raise $\sqrt[3]{x}$ to the power of $1$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (16 + 4 \sqrt[3]{x} + 4 \sqrt[3]{x} + \sqrt[3]{x} \sqrt[3]{x}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.5.1.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (16 + 4 \sqrt[3]{x} + 4 \sqrt[3]{x} + {\sqrt[3]{x}}^{1 + 1}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.5.1.3.3

Add $1$ and $1$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (16 + 4 \sqrt[3]{x} + 4 \sqrt[3]{x} + {\sqrt[3]{x}}^{2}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.5.1.4

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (16 + 4 \sqrt[3]{x} + 4 \sqrt[3]{x} + \sqrt[3]{x^{2}}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.5.2

Add $4 \sqrt[3]{x}$ and $4 \sqrt[3]{x}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 (16 + 8 \sqrt[3]{x} + \sqrt[3]{x^{2}}) + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.6

Apply the distributive property.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 12 \cdot 16 - 12 (8 \sqrt[3]{x}) - 12 \sqrt[3]{x^{2}} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.7

Simplify.

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Step 5.2.3.7.1

Multiply $- 12$ by $16$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 192 - 12 (8 \sqrt[3]{x}) - 12 \sqrt[3]{x^{2}} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.7.2

Multiply $8$ by $- 12$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 192 - 96 \sqrt[3]{x} - 12 \sqrt[3]{x^{2}} + 48 (4 + \sqrt[3]{x}) - 64$

Step 5.2.3.8

Apply the distributive property.

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 192 - 96 \sqrt[3]{x} - 12 \sqrt[3]{x^{2}} + 48 \cdot 4 + 48 \sqrt[3]{x} - 64$

Step 5.2.3.9

Multiply $48$ by $4$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 192 - 96 \sqrt[3]{x} - 12 \sqrt[3]{x^{2}} + 192 + 48 \sqrt[3]{x} - 64$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $64 + 48 \sqrt[3]{x} + 12 \sqrt[3]{x^{2}} + x - 192 - 96 \sqrt[3]{x} - 12 \sqrt[3]{x^{2}} + 192 + 48 \sqrt[3]{x} - 64$ .

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Step 5.2.4.1.1

Subtract $12 \sqrt[3]{x^{2}}$ from $12 \sqrt[3]{x^{2}}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + 0 + x - 192 - 96 \sqrt[3]{x} + 192 + 48 \sqrt[3]{x} - 64$

Step 5.2.4.1.2

Add $64 + 48 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + x - 192 - 96 \sqrt[3]{x} + 192 + 48 \sqrt[3]{x} - 64$

Step 5.2.4.1.3

Add $- 192$ and $192$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + x + 0 - 96 \sqrt[3]{x} + 48 \sqrt[3]{x} - 64$

Step 5.2.4.1.4

Add $64 + 48 \sqrt[3]{x} + x$ and $0$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 64 + 48 \sqrt[3]{x} + x - 96 \sqrt[3]{x} + 48 \sqrt[3]{x} - 64$

Step 5.2.4.1.5

Subtract $64$ from $64$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 0 + 48 \sqrt[3]{x} + x - 96 \sqrt[3]{x} + 48 \sqrt[3]{x}$

Step 5.2.4.1.6

Add $0$ and $48 \sqrt[3]{x}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = 48 \sqrt[3]{x} + x - 96 \sqrt[3]{x} + 48 \sqrt[3]{x}$

Step 5.2.4.2

Subtract $96 \sqrt[3]{x}$ from $48 \sqrt[3]{x}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = x - 48 \sqrt[3]{x} + 48 \sqrt[3]{x}$

Step 5.2.4.3

Combine the opposite terms in $x - 48 \sqrt[3]{x} + 48 \sqrt[3]{x}$ .

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Step 5.2.4.3.1

Add $- 48 \sqrt[3]{x}$ and $48 \sqrt[3]{x}$ .

$f^{- 1} (4 + \sqrt[3]{x}) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (4 + \sqrt[3]{x}) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (x^{3} - 12 x^{2} + 48 x - 64)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} - 12 x^{2} + 48 x - 64) = 4 + \sqrt[3]{x^{3} - 12 x^{2} + 48 x - 64}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Make each term match the terms from the binomial theorem formula.

$f (x^{3} - 12 x^{2} + 48 x - 64) = 4 + \sqrt[3]{x^{3} - 3 \cdot (4 x^{2}) + 3 \cdot (4^{2} x) - 1 \cdot 4^{3}}$

Step 5.3.3.2

Factor $x^{3} - 3 \cdot 4 x^{2} + 3 \cdot 4^{2} x - 1 \cdot 4^{3}$ using the binomial theorem.

$f (x^{3} - 12 x^{2} + 48 x - 64) = 4 + \sqrt[3]{{(x - 4)}^{3}}$

Step 5.3.3.3

Pull terms out from under the radical, assuming real numbers.

$f (x^{3} - 12 x^{2} + 48 x - 64) = 4 + x - 4$

Step 5.3.4

Combine the opposite terms in $4 + x - 4$ .

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Step 5.3.4.1

Subtract $4$ from $4$ .

$f (x^{3} - 12 x^{2} + 48 x - 64) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (x^{3} - 12 x^{2} + 48 x - 64) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} - 12 x^{2} + 48 x - 64$ is the inverse of $f (x) = 4 + \sqrt[3]{x}$ .

$f^{- 1} (x) = x^{3} - 12 x^{2} + 48 x - 64$